/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 Assume that relative maximum and... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 19

Assume that relative maximum and minimum values are absolute maximum and minimum values. ProHauling Services is designing an open-top, rectangular container that will have a volume of \(320 \mathrm{ft}^{3}\). The cost of making the bottom of the container is \(\$ 5\) per square foot, and the cost of the sides is \(\$ 4\) per square foot. Find the dimensions of the container that will minimize total cost. (Hint: Make a substitution using the formula for volume.)

Short Answer

Expert verified
The optimal dimensions are 8 ft by 8 ft by 5 ft.

Step by step solution

01

Identify Variables and Costs

Let's define the dimensions of the container. Let \( x \) be the width, \( y \) be the length, and \( h \) be the height of the container. The container's volume is given by the equation \( V = x \, y \, h = 320 \; \text{ft}^3 \). The cost for the bottom, which has area \( x \, y \), is \( 5 \times x \, y \). The total side wall area consists of \( 2xh + 2yh \), leading to a cost of \( 4 \times (2xh + 2yh) \). Therefore, the total cost \( C \) is \( 5xy + 8(xh + yh) \).
02

Express Cost as a Function of Two Variables

It's beneficial to express \( h \) in terms of \( x \) and \( y \) using the volume equation: \( h = \frac{320}{xy} \). Substituting \( h \) into the cost equation gives us \( C = 5xy + 8 \left(x \frac{320}{xy} + y \frac{320}{xy}\right) \). Simplifying this gives \( C = 5xy + \frac{2560}{x} + \frac{2560}{y} \). This function is now in terms of \( x \) and \( y \).
03

Substitute One Variable Using Symmetry

To simplify further, consider the case where the container is symmetric in terms of width and length. Without loss of generality, assume \( x = y \). Then \( xy = x^2 = 320 \, h = \frac{320}{x^2} \). Substituting into the cost equation, we have \( C = 5x^2 + 2 \left(\frac{2560}{x}\right) = 5x^2 + 5120/x \).
04

Differentiate to Find Critical Points

Take the derivative of \( C \) with respect to \( x \): \( \frac{dC}{dx} = 10x - \frac{5120}{x^2} \). To find critical points, set the derivative equal to zero: \( 10x - \frac{5120}{x^2} = 0 \). Solve for \( x \) to find \( x^3 = 512 \) or \( x = 8 \).
05

Verify with Second Derivative Test

Calculate the second derivative: \( \frac{d^2C}{dx^2} = 10 + \frac{10240}{x^3} \). Evaluate it at \( x = 8 \): \( \frac{d^2C}{dx^2} = 10 + \frac{10240}{512} \), which is positive, indicating a local minimum at \( x = 8 \).
06

Find Corresponding Dimensions

Given \( x = 8 \), \( y = 8 \). Calculate \( h = \frac{320}{64} = 5 \). Thus, the dimensions of the container are 8 ft by 8 ft by 5 ft.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative Applications
Derivative applications are essential in calculus to optimize functions, often to find maximum or minimum values. When you need to design something efficiently 鈥� like a container with minimum cost 鈥� derivatives become key tools. For our container problem, the goal is to minimize the total cost of materials used by adjusting dimensions. We express the cost as a function, in terms of the container's dimensions, including height, width, and length. Derivatives help in identifying where this cost function reaches its lowest point.
  • First Derivative: It shows how the function behaves, indicating whether it's increasing or decreasing at a certain point.
  • Setting the Derivative to Zero: This step finds the critical points, where the cost could potentially be minimized or maximized.
  • Second Derivative Test: Confirm whether these critical points are indeed minimums by checking the concavity. If the second derivative is positive at the point, the graph is concave up, indicating a minimum.
In our exercise, deriving and simplifying the cost function helped us find the point where the cost was minimized, with a preference for symmetry in the box's width and length.
Volume in Geometry
Volume calculations are crucial in geometry, as they help determine the space that an object occupies. In our problem, we need to maintain a specific volume while minimizing costs. The volume equation for our rectangular container is given by \( V = x \, y \, h \).
  • Understanding Volume Formula: The volume formula is the product of the dimensions: height, width, and length. For our open-top container, keeping the volume fixed at 320 cubic feet constrains other variables.
  • Substitution for Simplification: We can express one variable in terms of the others using the volume formula, such as substituting for height as \( h = \frac{320}{xy} \). This helps reduce the number of variables.
  • Symmetry and Design:** In optimization, symmetric designs often lead to easier calculations and aesthetically pleasing results. Here, assuming equal width and length simplifies finding the dimensions that minimize costs.
Thus, understanding the volume's geometric importance shapes how we approach minimizing other attributes, like cost, efficiently.
Cost Minimization
Cost minimization focuses on reducing expenses while fulfilling requirements, a common goal in design and engineering problems. In this case, it's about creating a cost-effective container with a fixed volume.
  • Breaking Down Costs: The costs are divided into the bottom and sides of the container, with each having a different cost per square foot. Calculating these separately helps in forming a comprehensive cost function.
  • Formulating the Cost Function: By adding the costs of each component, the total cost function is written as \( C = 5xy + 8(xh + yh) \), eventually simplifying to \( C = 5xy + \frac{2560}{x} + \frac{2560}{y} \).
  • Optimization Solutions: Setting the derivative of this cost function to zero gives critical points that potentially provide the minimum cost solution. Here, symmetry (i.e., \( x = y \)) further reduces complexity.
Minimizing cost within specified constraints requires a strategic approach, combining algebraic manipulation with calculus tools to achieve the optimal solution efficiently.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider \(f(x, y)=x^{3}-5 x y^{2}\). Show that \(f\) is a solution of the partial differential equation $$x f_{x y}-f_{y}=0$$

Evaluate. $$ \int_{1}^{3} \int_{0}^{x} 2 e^{x^{2}} d y d x $$

Find \(f_{x}, f_{y},\) and \(f_{\lambda}\). $$f(x, y, \lambda)=9 x y-\lambda(3 x-y+7)$$

From the center of Bridgeton, a water main runs northwest at a \(45^{\circ}\) angle relative to due north and due west. A new house is being built at a point 2 miles west and 1 mile north of the center of town. The cost to connect the house to the water main is \(\$ 8,000\) per mile. What is the shortest possible distance (to three decimal places) from the house to the water main, and what would be the minimum cost, to the nearest dollar, of connecting the house to the water main?

Find the volume of the solid capped by the surface \(z=x+y\) over the region bounded on the \(x y\) -plane by \(y=1-x, y=0, x=0,\) and \(x=1,\) by evaluating the integral $$ \int_{0}^{1} \int_{0}^{1-x}(x+y) d y d x $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.